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<span id="sphx-glr-auto-examples-neighbors-plot-nca-dim-reduction-py"></span><h1>Dimensionality Reduction with Neighborhood Components Analysis<a class="headerlink" href="#dimensionality-reduction-with-neighborhood-components-analysis" title="Permalink to this headline">¶</a></h1>
<p>Sample usage of Neighborhood Components Analysis for dimensionality reduction.</p>
<p>This example compares different (linear) dimensionality reduction methods
applied on the Digits data set. The data set contains images of digits from
0 to 9 with approximately 180 samples of each class. Each image is of
dimension 8x8 = 64, and is reduced to a two-dimensional data point.</p>
<p>Principal Component Analysis (PCA) applied to this data identifies the
combination of attributes (principal components, or directions in the
feature space) that account for the most variance in the data. Here we
plot the different samples on the 2 first principal components.</p>
<p>Linear Discriminant Analysis (LDA) tries to identify attributes that
account for the most variance <em>between classes</em>. In particular,
LDA, in contrast to PCA, is a supervised method, using known class labels.</p>
<p>Neighborhood Components Analysis (NCA) tries to find a feature space such
that a stochastic nearest neighbor algorithm will give the best accuracy.
Like LDA, it is a supervised method.</p>
<p>One can see that NCA enforces a clustering of the data that is visually
meaningful despite the large reduction in dimension.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1"># License: BSD 3 clause</span>

<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">datasets</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
<span class="kn">from</span> <span class="nn">sklearn.decomposition</span> <span class="kn">import</span> <span class="n">PCA</span>
<span class="kn">from</span> <span class="nn">sklearn.discriminant_analysis</span> <span class="kn">import</span> <span class="n">LinearDiscriminantAnalysis</span>
<span class="kn">from</span> <span class="nn">sklearn.neighbors</span> <span class="kn">import</span> <span class="p">(</span><span class="n">KNeighborsClassifier</span><span class="p">,</span>
                               <span class="n">NeighborhoodComponentsAnalysis</span><span class="p">)</span>
<span class="kn">from</span> <span class="nn">sklearn.pipeline</span> <span class="kn">import</span> <span class="n">make_pipeline</span>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span>

<span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>

<span class="n">n_neighbors</span> <span class="o">=</span> <span class="mi">3</span>
<span class="n">random_state</span> <span class="o">=</span> <span class="mi">0</span>

<span class="c1"># Load Digits dataset</span>
<span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">datasets</span><span class="o">.</span><span class="n">load_digits</span><span class="p">(</span><span class="n">return_X_y</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>

<span class="c1"># Split into train/test</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> \
    <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">stratify</span><span class="o">=</span><span class="n">y</span><span class="p">,</span>
                     <span class="n">random_state</span><span class="o">=</span><span class="n">random_state</span><span class="p">)</span>

<span class="n">dim</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">n_classes</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">unique</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>

<span class="c1"># Reduce dimension to 2 with PCA</span>
<span class="n">pca</span> <span class="o">=</span> <span class="n">make_pipeline</span><span class="p">(</span><span class="n">StandardScaler</span><span class="p">(),</span>
                    <span class="n">PCA</span><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="n">random_state</span><span class="p">))</span>

<span class="c1"># Reduce dimension to 2 with LinearDiscriminantAnalysis</span>
<span class="n">lda</span> <span class="o">=</span> <span class="n">make_pipeline</span><span class="p">(</span><span class="n">StandardScaler</span><span class="p">(),</span>
                    <span class="n">LinearDiscriminantAnalysis</span><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="mi">2</span><span class="p">))</span>

<span class="c1"># Reduce dimension to 2 with NeighborhoodComponentAnalysis</span>
<span class="n">nca</span> <span class="o">=</span> <span class="n">make_pipeline</span><span class="p">(</span><span class="n">StandardScaler</span><span class="p">(),</span>
                    <span class="n">NeighborhoodComponentsAnalysis</span><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
                                                   <span class="n">random_state</span><span class="o">=</span><span class="n">random_state</span><span class="p">))</span>

<span class="c1"># Use a nearest neighbor classifier to evaluate the methods</span>
<span class="n">knn</span> <span class="o">=</span> <span class="n">KNeighborsClassifier</span><span class="p">(</span><span class="n">n_neighbors</span><span class="o">=</span><span class="n">n_neighbors</span><span class="p">)</span>

<span class="c1"># Make a list of the methods to be compared</span>
<span class="n">dim_reduction_methods</span> <span class="o">=</span> <span class="p">[(</span><span class="s1">&#39;PCA&#39;</span><span class="p">,</span> <span class="n">pca</span><span class="p">),</span> <span class="p">(</span><span class="s1">&#39;LDA&#39;</span><span class="p">,</span> <span class="n">lda</span><span class="p">),</span> <span class="p">(</span><span class="s1">&#39;NCA&#39;</span><span class="p">,</span> <span class="n">nca</span><span class="p">)]</span>

<span class="c1"># plt.figure()</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">model</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">dim_reduction_methods</span><span class="p">):</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
    <span class="c1"># plt.subplot(1, 3, i + 1, aspect=1)</span>

    <span class="c1"># Fit the method&#39;s model</span>
    <span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>

    <span class="c1"># Fit a nearest neighbor classifier on the embedded training set</span>
    <span class="n">knn</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">),</span> <span class="n">y_train</span><span class="p">)</span>

    <span class="c1"># Compute the nearest neighbor accuracy on the embedded test set</span>
    <span class="n">acc_knn</span> <span class="o">=</span> <span class="n">knn</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">),</span> <span class="n">y_test</span><span class="p">)</span>

    <span class="c1"># Embed the data set in 2 dimensions using the fitted model</span>
    <span class="n">X_embedded</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>

    <span class="c1"># Plot the projected points and show the evaluation score</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X_embedded</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X_embedded</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;Set1&#39;</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;</span><span class="si">{}</span><span class="s2">, KNN (k=</span><span class="si">{}</span><span class="s2">)</span><span class="se">\n</span><span class="s2">Test accuracy = </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">name</span><span class="p">,</span>
                                                              <span class="n">n_neighbors</span><span class="p">,</span>
                                                              <span class="n">acc_knn</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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  // Returns true when headerlink target matches hash in url
  (function() {
    hashTargetOnTop = function() {
        var hash = window.location.hash;
        if ( hash.length < 2 ) { return false; }

        var target = document.getElementById( hash.slice(1) );
        if ( target === null ) { return false; }

        var top = target.getBoundingClientRect().top;
        return (top < 2) && (top > -2);
    };

    // Hide navbar on load if hash target is on top
    var navBar = document.getElementById("navbar");
    var navBarToggler = document.getElementById("sk-navbar-toggler");
    var navBarHeightHidden = "-" + navBar.getBoundingClientRect().height + "px";
    var $window = $(window);

    hideNavBar = function() {
        navBar.style.top = navBarHeightHidden;
    };

    showNavBar = function() {
        navBar.style.top = "0";
    }

    if (hashTargetOnTop()) {
        hideNavBar()
    }

    var prevScrollpos = window.pageYOffset;
    hideOnScroll = function(lastScrollTop) {
        if (($window.width() < 768) && (navBarToggler.getAttribute("aria-expanded") === 'true')) {
            return;
        }
        if (lastScrollTop > 2 && (prevScrollpos <= lastScrollTop) || hashTargetOnTop()){
            hideNavBar()
        } else {
            showNavBar()
        }
        prevScrollpos = lastScrollTop;
    };

    /*** high preformance scroll event listener***/
    var raf = window.requestAnimationFrame ||
        window.webkitRequestAnimationFrame ||
        window.mozRequestAnimationFrame ||
        window.msRequestAnimationFrame ||
        window.oRequestAnimationFrame;
    var lastScrollTop = $window.scrollTop();

    if (raf) {
        loop();
    }

    function loop() {
        var scrollTop = $window.scrollTop();
        if (lastScrollTop === scrollTop) {
            raf(loop);
            return;
        } else {
            lastScrollTop = scrollTop;
            hideOnScroll(lastScrollTop);
            raf(loop);
        }
    }
  })();
});

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